Before you critique something, you should probably know what it means. They talked about Sony going bankrupt, which is very different from going out of business. Sony has 15 billion dollars or so in debt, and they're not exactly raking in the dough right now. If they can't pay their creditors, they go bankrupt.
Bankruptcy does not mean that Sony would cease operating. In the US, there are basically two types of bankruptcy. There is Chapter 7 and 11. Chapter 7 bankruptcy means that Sony would have to sell off all their assets (Playstation Brand, vaio brand, any buildings, patents, w/e) and pay what is possible to their creditors. On the other hand, Chapter 11 would mean that Sony would work with their creditors and the government to come up with a plan that would enable them to stay operational, and arrange a timetable by which creditors would be repayed. Under this program, creditors and government might have some authority over how Sony can run their business. If Sony is eventually able to pay their debts, they'd be able to come out of bankruptcy.
Sony's main problem is their TV division. The problem is (in my not so informed opinion) two fold. First off, TVs were once the only screen in the house which all your entertainment flowed through. So, the advent of smart phones and computer based streaming has diminished the importance of TVs. More importantly, most people have already made the switch to HD, and the prices of HD TVs fell rather quickly. TV companies are working hard to create the next reason to upgrade (HD, OLED, and 4K oh my!) but people have been reluctant to upgrade. Companies like Samsung and LG have also become big players in the TV field and have knocked Sony down on the TV ladder.
It would be easy to say "oh well then just stop making TVs", but that isn't really an option. TVs are high profit items, and provide the best opportunity for Sony to make the kind of money that they need to repay their creditors. Even if the PS4 is wildly successful, that isn't going to generate the revenue they need. They would basically need to make as much money as Nintendo since 2006 off gaming. Considering weaker first party software sales, and a failing portable, there is no way Sony can do that with the PS4 alone.
So it seems to me that Sony has to improve their cell phone division, or they need to convince people to upgrade to 4K tvs. If they can't do one of these two things, then being bankrupt in 2 years is a very real possibility. I mean, they have 15 billion dollars in debt and they expect to make 300 million dollars this year. They obviously don't have to pay all their debt in that time frame, but still isn't a good thing. Something's gotta give.
As for Nintendo, that one seems a bit off. Nintendo has basically no debt, and 14 million in cash/liquid assets. So, it's hard to see them burning through that cash reserve in 2 years time. They're a pretty conservative company that doesn't tend to take huge risks.
That being said, technology is volatile, and stranger things have happened in 2 years. Suppose a company like Apple made a serious portable game machine targeting the DS market directly. Not saying that would succeed, but suppose they did it right. Point is, things can change real quick.